## Module 6 – General Linear Models: ANOVA & ANCOVA (2015)

Module 6 – General Linear Models: ANOVA & ANCOVAThe two exercises below utilize the data sets career-a.sav and career-f.sav, which can be downloaded from this Web site:.Pyrczak.com/data”>www.Pyrczak.com/data1. You are interested in evaluating the effect of job satisfaction (satjob2) and age category (agecat4) on the combined DV of hours worked per week (hrsl) and years of education (educ). Use career-a.sav for steps a and b.a. Develop the appropriate research questions and/or hypotheses for main effects and interaction.b. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing data and outliers?For all subsequent analyses in Question 1, use career-f.sav and the transformed variables of hrs2 and educ 2.c. Test the assumptions of normality and linearity of DVs.i. What steps, if any, are necessary for increasing normality?ii. Are DVs linearly related?d. Conduct MANOVA with post hoc (be sure to test for homogeneity of variance-covariance).a. Can you conclude homogeneity of variance-covariance? Which test statistic is most apÂ¬propriate for interpretation of multivariate results?b. Is factor interaction significant? Explain.c. Are main effects significant? Explain.d. What can you conclude from univariate ANOVA and post hoc results?e. Write a results statement.2. Building on the previous problem, in which you investigated the effects of job satisfaction (satjobl) and age category (agecat4) on the combined dependent variable of hours worked per week (hrsl) and years of education (educ), you are now interested in controlling for respondents’ income such that rin- com91 will be used as a covariate. Complete the following using career-a.sav.i. Develop the appropriate research questions and/or hypotheses for main effects and interaction.ii. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing data and outliers?For all subsequent analyses in Question 2, use career-f.sav and the transformed variables of hrs2, educ2, and rincoml.iii. Test the assumptions of normality and linearity of DVs and covariate.i. What steps, if any, are necessary for increasing normality?ii. Are DVs and covariate linearly related?c. Conduct a preliminary MANCOVA to test the assumptions of homogeneity of variance- covariance and homogeneity of regression slopes/planes.i. Can you conclude homogeneity of variance-covariance? Which test statistic is most appropriate for interpretation of multivariate results?ii. Do factors and covariate significantly interact? Explain.d. Conduct MANCOVA.i. Is factor interaction significant? Explain.ii. Are main effects significant? Explain.iii. What can you conclude from univariate ANOVA results?e. Write a results statement.3. Compare the results from Question 1 and Question 2. Explain the differences in main effects.The following output was generated from conducting a forward multiple regression to identify which IVs {urban, birthrat, Inphone, and Inradio) predict Ingdp. The data analyzed were from the SPSS country-a.sav data file.Variables Entered/Removed 1 Variables Variables Model Entered Removed Method1 Forward (Criterion: LNPHONE Probability y-of-F-to-e nter <= .050)2 Forward (Criterion: BIRTHRAT Probability y-of-F-to-e nter <= 050)a Dependent Variable: LNGDPModel Summary Std. Error Change Statistics Adjusted of the R Square Sig. FModel R R Square R Square Estimate Change F Change df1 df2 Change1 ,941a .886 .885 5180 886 862 968 1 111 ,0002 .943" .890 888 .5109 .004 4.095 1 110 .045a Predictors: (Constant), LNPHONE b Predictors: (Constant), LNPHONE, BIRTHRATCoefficients"Model Unstandardized Coefficients Standardized Coefficien ts t Sifl. Correlations Collinearity Statistics B Std Error Beta Zero-order Partial Part Tolerance VIF1 (Constant) LNPHONE 6 389 .736 058 .025 941 110 662 29.376 .000 000 .941 941 941 1.000 1.0002 (Constant) LNPHONE BIRTHRAT 6 878 663 -1 29E-02 .248 .044 006 849- 113 27 744 15238 -2 024 000 .000 045 941 -811 .824 - 189 482 -064 .322 322 3 104 3 104a Dependent Variable LNGDPa- Predictors: (Constant), LNPHONE b Predictors: (Constant), LNPHONE, BIRTHRAT c. Dependent Variable: LNGDPExcluded Variables Collinearitv Statistics Partial MinimumModel Beta In t Sip. Correlation Tolerance VIF Tolerance1 URBAN 095a 1.901 .060 .178 .404 2.475 404 BIRTHRAT -.113a -2.024 045 -.189 .322 3.104 .322 LN RADIO .026a .557 .579 .053 .461 2.171 .4612 URBAN ,091b 1.848 .067 .174 .403 2.479 .225 LNRADIO 021b .455 .650 044 .459 2.178 .243a Predictors in the Model: (Constant), LNPHONE b- Predictors in the Model: (Constant), LNPHONE. BIRTHRAT c. Dependent Variable: LNGDPANOVAcModel Sum of Squares df Mean Square F Siq.1 Regression Residual Total 231.539 29.782 261.321 1111 112 231.539 .268 862.968 ,000a2 Regression Residual Total 232.608 28.713 261.321 2 110 112 116.304 .261 445.561 .000"a. Predictors: (Constant), Inphoneb. Predictors: (Constant), Inphone, birthratc. Dependent Variable: Ingdpi. Evaluate the tolerance statistics. Is multicollinearity a problem .'ii. What variables create the model to predict Ingdp? What statistics support your response?iii. Is the model significant in predicting Ingdp? Explain.iv. What percentage of variance in Ingdp is explained by the model?v. Write the regression equation for Ingdp.This question utilizes the data sets profile-a.sav and profile-b.sav, which can be downloaded from this Web site:.Pvrczak.com/data">www.Pvrczak.com/dataYou are interested in examining whether the variables shown here in brackets [years of age (age), hours worked per week (hrsl), years of education (educ), years of education for mother (maeduc), and years of education for father (paeduc)] are predictors of individual income (rincmdol). Complete the following steps to conduct this analysis.a. Using profile-a.sav, conduct a preliminary regression to calculate Mahalanobis distance. IdenÂ¬tify the critical value for chi-square. Conduct Explore to identify outliers. Which cases should be removed from further analysis?For all subsequent analyses, use profile-b.saw Make sure that only cases where MAH l < 22.458 are selected.b. Create a scatterplot matrix. Can you assume linearity and normality?c. Conduct a preliminary regression to create a residual plot. Can you assume normality and ho- moscedasticity?d. Conduct multiple regression using the Enter method. Evaluate the tolerance statistics. Is multicollinearity a problem?e. Does the model significantly predict rincmdol? Explain.f. Which variables significantly predict rincmdol? Which variable is the best predictor of the DV?g. What percentage of variance in rincmdol is explained by the model?h. Write the regression equation for the standardized variables.i. Explain why the variables of mother's and father's education are not significant predictors of rincmdol.The following exercises seek to determine what underlying structure exists among the following variables in profile-a.sav: highest degree earned (degree), hours worked per week (hrsl), job satisfaction (satjob), years of education (educ), hours per day watching TV (tvhours), general happiness {happy), degree to which life is exciting(life), and degree to which the lot of the average person is getting worse (anomiaS).1. The following output was generated for the initial analysis. Varimax rotation wa$ utilized.Communalities Initial Extractiondegree 1 000 933hrs1 1 000 602satjob 1 000 447educ 1 000 939tvhours 1 000 556happy 1 000 .576life 1 000 500anomia5 1 000 317Extraction Method Principal Component AnalysisTotal Variance ExplainedComponent Initial Eigenvalues Extraction Sums of Squared loadings Rotation Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %1 2423 30 293 30 293 2 423 30 293 30 293 1 879 23 488 23 4882 1 426 17.822 48 115 1 426 17 822 48 115 1 734 21.676 45 1653 1 021 12 760 60 875 1 021 12 760 60 875 1 257 15.710 60 8754 886 11.077 71 952 5 796 9 955 81 907 6 728 9 094 91 001 7 .607 7 589 98 590 8 113 1 410 100 000 Extraction Method: Principal Component Analysis"n Metho Scree Plgf 9 plot Component NumberReproduced Correlations degree hrs1 satjob educ tvhours happy life anomia5Reproduced Correlation degree .933" .176 -.039 .935 -.239 -.119 .230 .118 hrs1 .176 ,602b -239 .194 -.576 -.077 .141 -.049 satjob -.039 -.239 447b -.062 .214 .469 -.436 -.297 educ .935 194 -062 939b -.255 -.142 .252 .131 tvhours -.239 -.576 .214 -.255 .556" .066 -.136 .047 happy -.119 -.077 .469 -.142 .066 576b -.526 -.412 life .230 .141 -.436 .252 -.136 -.526 500b .371 anomia5 .118 -.049 -.297 .131 .047 -.412 .371 ,317bResidualÂ® degree .004 -068 -.050 .032 -.004 -.034 -.037 hrs1 .004 .104 .011 .361 -.031 -.046 .112 satjob -.068 .104 -.037 -.105 -.197 .151 .158 educ -.050 .011 -.037 .026 -.002 -029 -.012 tvhours 032 .361 -.105 .026 .014 -.012 -.099 happy -.004 -.031 -.197 -.002 .014 .159 .177 life -.034 -.046 .151 -.029 -.012 .159 -.217 anomia5 -.037 .112 .158 -.012 -099 .177 -.217 Extraction Method: Principal Component Analysis.a Residuals are computed between observed and reproduced correlations. There are 12 (42.0%) nonredundant residuals with absolutevalues greater than 0.05.b. Reproduced communalitiesa. Assess the eigenvalue criterion. How many components were retained? Is the eigenvalue ap-propriate, considering the number of factors and the communalities?b. Assess the variance explained by the retained components. What is the total variability ex-plained by the model? Is this amount adequate?c. Assess the scree plot. At which component does the plot begin to level off?d. Assess the residuals. How many residuals exceed the .05 criterion?e. Having applied the four criteria, do you believe the number of components retained in this analyÂ¬sis is appropriate? If not, what is your recommendation?2. Assume that you believe four components should be retained from the analysis in the previous exerÂ¬cise. Conduct a factor analysis with varimax rotation (be sure to retain four components).a. Evaluate each of the four criteria. Has the model fit improved? Explain.b. Provide two alternatives for improving the model. Prediction and Association Practice ExerciseUse Practice Data Set 2 in Appendix B. If we want to predict salary from years of education, what salary would you predict for someone with 12 years of education? What salary would you predict for someone with a college education (16 years)?Use Practice Data Set 2 in Appendix B. Determine the prediction equation for preÂ¬dicting salary based on education, years of service, and sex. Which variables are significant predictors? If you believe that men were paid more than women were, what would you conclude after conducting this analysis?Data Set 2 Appendix BA survey of employees is conducted. Each employee provides the following inforÂ¬mation: Salary (SALARY), Years of Service (YOS), Sex (SEX), Job Classification (CLASSIFY), and Education Level (EDUC). Note that you will have to code SEX (Male = 1, Female = 2) and CLASSIFY (Clerical = 1, Technical = 2, Professional = 3), and indicate that they are measured on a nominal scale. Name A Jj?> [Jwidth “{“Decimals] Label Values Missin g ] Columns E Align Measure T~ Role1 salary Numeric 8 2 None None 8 9 Right f Scale S Input2 yos Numeric 8 2 None None 8 m Right # Scale Input3 sex Numeric 8 2 (1 00. Male) None 8 m Right Nominal Input4 classify Numeric 8 2 {1 00, Cleric None 8 9 Right A Nominal Input5 educ Numeric 8 2 None None 8 9 Right f Scale InputSALARY YOS SEX CLASSIFY EDUC35,000 8 Male Technical 1418,000 4 Female Clerical 1020,000 1 Male Professional 1650,000 20 Female Professional 1638,000 6 Male Professional 2020,000 6 Female Clerical 1275,000 17 Male Professional 2040,000 4 Female Technical 1230,000 8 Male Technical 1422,000 15 Female Clerical 1223,000 16 Male Clerical 1245,000 2 Female Professional 16

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